Between 10,000 and 99,999, there are 90,000 five-digit integers. Out of that 90,000 there are exactly 24,000 integers that are NOT divisible by any number in the set of {1, 2, 3, 4, 5}, with the exception, of course, of being divisible by 1.
Therefore, the probability is: 24,000 / 90,000 = 4 / 15